How do you rationalize the denominator and simplify #(4+sqrt3)/(5+sqrt2)#?

Answer 1

To rationalize the denominator we must get rid of any radicals in the denominator.

We can rationalize the denominator of a binomial by multiplying the numerator and the denominator by the conjugate of the denominator.

Definition: the conjugate is what forms a difference of squares. Ex the conjugate of #(a + b)# is #(a - b)#, since #(a + b)(a - b) = a^2 - b^2#. Essentially, to find the conjugate all you have to do is simply to switch the sign #+ or -# sign between the two terms in the binomial.
Thus, the conjugate of #5 + sqrt(2)# is #5 - sqrt(2)#.
#(4 + sqrt(3))/(5 + sqrt(2)) xx (5 - sqrt(2))/(5 - sqrt(2))#
Don't forget that you cannot multiply non radicals with radicals. E.g #3 xx sqrt(8) != sqrt(24)# but is simply #3sqrt(8)#.
#-> (20 + 5sqrt(3) - 4sqrt(2) - sqrt(6))/(25 + 2sqrt(5) - 2sqrt(5) - sqrt(4)#
= #(20 + 5sqrt(3) - 4sqrt(2) - sqrt(6))/23#

Practice exercises:

Rationalize the denominator of #(3sqrt(5) - 2sqrt(7))/(2sqrt(3) - 4sqrt(2))#

Good luck!

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Answer 2

To rationalize the denominator and simplify (4+sqrt3)/(5+sqrt2), we multiply both the numerator and denominator by the conjugate of the denominator, which is (5-sqrt2). This results in (4+sqrt3)(5-sqrt2) in the numerator and (5+sqrt2)(5-sqrt2) in the denominator. Expanding and simplifying these expressions gives (20-4sqrt2+5sqrt3-sqrt6) in the numerator and (25-2) in the denominator. Combining like terms, the simplified expression is (18+5sqrt3-4sqrt2-sqrt6)/23.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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